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This paper presents some of the author's experimental results in applying a family of iterative methods, the family of EN-like methods Eirola & Nevanlinna (1989), to equations obtained from the discretization of the nonlinear two dimentional Poisson equation occuring in semiconductor device modelling. It is shown that these iterative methods are efficient both in computation times and in storage requirements in comparison with other known methods. [ABSTRACT FROM AUTHOR]

In this article, we study the properties of the hyperinterpolation operator on the unit disc D in ℝ2. We show how hyperinterpolation can be used in connection with the Kumar–Sloan method to approximate the solution of a nonlinear Poisson equation on the unit disc (discrete Galerkin method). A bound for the norm of the hyperinterpolation operator in the space C(D) is derived. Our results prove the convergence of the discrete Galerkin method in the maximum norm if the solution of the Poisson equation is in the class C1, δ(D), δ > 0. Finally, we present numerical examples which show that the discrete Galerkin method converges faster than O(n–k), for every k ∈ ℕ if the solution of the nonlinear Poisson equation is in C∞(D). [ABSTRACT FROM PUBLISHER]

This paper applies radial point interpolation collocation method (RPICM) for solving nonlinear Poisson equations arising in computational chemistry and physics. Thin plate spline (TPS) Radial basis functions are used in the work. A series of test examples are numerically analysed using the present method, including 2D Liouville equation, Bratu problem and Poisson-Boltzmann equation, in order to test the accuracy and efficiency of the proposed schemes. Several aspects have been numerically investigated, namely the enforcement of additional polynomial terms; and the application of the Hermite-type interpolation which makes use of the normal gradient on Neumann boundary for the solution of PDEs with Neumann boundary conditions. Particular emphasis was on an efficient scheme, namely Hermite-type interpolation for dealing with Neumann boundary conditions. The numerical results demonstrate that a good accuracy can be obtained. The h-convergence rates are also studied for RPICM with coarse and fine discretization models. [ABSTRACT FROM AUTHOR]

Abstract: Numerical schemes for approximating the inverse of a given matrix, using an ordinary differential equation (ODE) model, were recently developed. We extend that approach to solve nonlinear minimization problems, within the framework of a preconditioned gradient method. The main idea is to develop an automatic and implicit scheme to approximate directly the preconditioned search direction at every iteration, without an a priori knowledge of the Hessian of the objective function, and involving only a reduced and controlled amount of storage and computational cost. The new scheme allows us to obtain asymptotically the Newton''s direction by improving the accuracy in the ODE solver associated with the implicit scheme. We will present extensive and encouraging numerical results on some well-known test problems, on the problem of computing the square root of a given symmetric and positive definite matrix, and also on a nonlinear Poisson type equation. [Copyright &y& Elsevier]